A couple of brief remarks before proceeding. First, in what follows we assume that the programs translating aggregation rules do not modify the truth value of the propositional variables in Bn,m. Hence, we exclude all programs like maj∗, expressing the majority rule, presented in Chapter 2. However, we could recover them by using some additional propositonal variables to store the initial profile and then execute the program for ruleF.
Secondly, notice that we could have defined an aggregation rule F as a function takingany profile, and not just the rational ones, as input. In this case, we would not have to check that the agents satisfy IC in their individual ballots and we could replace program profIC(Bn,m,Om) with init(Om) from the translation of all the axioms and from
the formula introduced above, thus obtaining|= [init(Om) ;f(
Bn,m)]V1≤i≤kAi.
3.2
Single-profile Axioms
In this section we present axioms that make reference to just one profile at a time. This means that to check whether an aggregation ruleF satisfies a single-profile axiom it is sufficient to inspect the outcome of the rule on some profile and the profile itself.
Since in our framework profiles correspond to valuations, and since for single-profile axioms we do not have to make reference to different valuations (corresponding to different profiles), it is possible to express them as simple propositional formulas. Therefore, in order to express Unanimity, Issue-Neutrality, Domain-Neutrality and N-Monotonicity we will not need to use the modal part of our language.
Unanimity
A rule F isunanimous if in case all agents agree on some issue j, the outcome of F for issuej agrees with them. Namely, issuej should be collectively accepted if all the agents accept it, and it should be rejected if they all reject it in their individual ballots. This axiom can also be separated into two different versions, depending on whether all agents accept or rejectj: we speak of positive or negative unanimity, respectively.
Formally, the Unanimity axiom states that for rule F:
U : For any B, for all j ∈ I and for x∈ {0,1}, if bij =xfor all i∈ N then F(B)j =x.
We can express Unanimity as a propositional formula in the following way:
U := ^ j∈I ((^ i∈N pij)→pj)∧(( ^ i∈N ¬pij)→ ¬pj) .
The first big conjunction ensures that what follows is going to hold for any issue. Inside the parentheses, the two conjuncts deal with the positive and the negative version of unanimity. Now we can show that our translation of the axiom is correct with the proof of the proposition below.
Proposition 15. Let N = {1, . . . , n} and I = {1, . . . , m} be the sets of agents and issues, respectively. Let Bn,m be the set of propositional variables encoding the opinions of agents in N on issues in I. Let f be a DL-PA program translating some rule F. Then,
U holds ⇐⇒ |= [profIC(Bn,m,Om) ;f(Bn,m)]U.
Proof. For the left-to-right direction, consider an arbitrary valuation v and an arbitrary valuation v′ such that (v, v′) ∈ ∥profIC(Bn,m,
Om) ;f(Bn,m)∥. We want to show that
v′ ∈ ∥U∥, or more precisely that:
v′ ∈ ∥^ j∈I ((^ i∈N pij)→pj)∧(( ^ i∈N ¬pij)→ ¬pj) ∥.
Suppose, for reductio, that v′ ̸∈ ∥U∥. Given the definition of the interpretation, this means that there is somej ∈ I such thatv′ ̸∈ ∥((V
i∈N pij)→pj)∧((Vi∈N ¬pij)→ ¬pj)∥.
Assume, without loss of generality, that v′ ̸∈ ∥(V
i∈Npij) →pj)∥ is the case (a similar
reasoning can be done in the other case). Hence, we have that v′ ∈ ∥V
i∈N pij ∥ and
v′ ̸∈ ∥pj∥.
Since (v, v′) ∈ ∥profIC(Bn,m,
Om) ;f(Bn,m)∥, we have that v corresponds to some
rational profileB and v′ corresponds to F(B). By assumption, we have that F satisfies (U): in particular, this means that for all j ∈ I, if for all i ∈ N we have bij = 1, then
F(B)j = 1.
Recall that we assumed that v and v′ do not differ on the propositional variables in
Bn,m. Hence, the fact that v′ ∈ ∥Vi∈Npij∥ implies that v ∈ ∥Vi∈Npij∥. Therefore in
profileBwe have thatbij = 1 for alli∈ N, which implies thatF(B)j = 1. On the other
hand, this contradicts the fact thatv′ corresponds to F(B) and that v′ ̸∈ ∥pj∥. Hence,
v′ ∈ ∥U∥.
For the right-to-left direction, take an arbitrary rational profile B. We want to show that for any j ∈ I, if bij = 1 for all i∈ N, then F(B)j = 1 (and analogously for bij = 0,
so we can focus on this case without loss of generality). Suppose, for reductio, that this is not the case. Hence, for somej ∈ I in profile B, we have bij = 1 for all i ∈ N and
3.2 Single-profile Axioms 43
Consider now valuationsv and v′ corresponding to B and F(B) respectively. This means that (v, v′)∈ ∥profIC(Bn,m,
Om) ;f(Bn,m)∥ and that v′ ∈ ∥U∥. Since v and v′ do
not differ on the propositional variables in Bn,m, by spelling out the definition of the
interpretation we get that v′ ∈ ∥pj∥. This contradicts the fact that v′ corresponds to
F(B) and that F(B)j = 0. Therefore, we have that F(B) = 1.
Issue-Neutrality
A rule F is neutral with respect to the issues if, whenever two issues are treated in the same way in the input, they are treated in the same way in the output. That is, if the agents express the very same pattern of acceptance or rejection for the issues, the issues should either be both collectively accepted or rejected.
Formally, we have that for rule F:
NI : For any two issuesj, k ∈ I and any profile B, if for all i∈ N bij =bik then F(B)j =F(B)k.
A translation of Issue-Neutrality as a propositional formula is then the following:
NI := ^ j∈I ^ k∈I (^ i∈N (pij ↔pik))→(pj ↔pk) .
With the first two big conjunctions we can quantify over all possible choices of two issues. The conditional inside the parentheses makes sure that if the agents make the same decisions on the two issues, the output for them will be the same.
Proposition 16. Let N = {1, . . . , n} and I = {1, . . . , m} be the sets of agents and issues, respectively. Let Bn,m be the set of propositional variables encoding the opinions
of agents in N on issues in I. Let f be a DL-PA program translating some rule F. Then,
NI holds ⇐⇒ |= [profIC(Bn,m,Om) ;f(Bn,m)]NI.
Domain-Neutrality
An aggregation rule is neutral with respect to the domain if, whenever two issues are treated in an opposite way in the input, their output should be opposite. Namely, in case there is an opposite pattern of acceptance or rejection from the agents, one of the issues should be accepted in the outcome and the other rejected.
ND : For any two issuesj, k ∈ I and any profile B, if for all i∈ N bij = 1−bik then F(B)j = 1−F(B)k.
A propositional formula expressing Domain-Neutrality is the following:
ND := ^ j∈I ^ k∈I (^ i∈N (pij ↔ ¬pik))→(pj ↔ ¬pk) .
In an analogous way to how we expressed Issue-Neutrality, we first quantify over all the possible choices of two issues. Then, the implication inside the parentheses makes sure that if the issues are treated in an opposite way in the profile, the outcome for them should be opposite.
Proposition 17. Let N = {1, . . . , n} and I = {1, . . . , m} be the sets of agents and issues, respectively. Let Bn,m be the set of propositional variables encoding the opinions of agents in N on issues in I. Let f be a DL-PA program translating some rule F. Then,
ND holds ⇐⇒ |= [profIC(Bn,m,Om) ;f(Bn,m)]ND.
N-Monotonicity
An aggregation rule is neutral-monotonic if, whenever two issues j and k are such that if an agent in the profile accepts j then she also accepts k, and if there is some agent
s that rejects j and accepts k, if j is accepted in the outcome then also k must be accepted in the outcome. The intuitive idea of monotonicity captured here is thus one which compares the pattern of acceptance between two issues within the same profile: we will see later a different axiom of monotonicity that instead compares the pattern of acceptance of the same issuebetween two profiles. More formally, we have that the following has to hold for ruleF:
MN := For any two j, k ∈ I and any B, if b
ij = 1 implies bik = 1 for alli∈ N
and there iss∈ N such thatbsj = 0 and bsk = 1, then F(B)j = 1 implies F(B)k = 1.
A propositional formula expressing N-Monotonicity is as follows:
MN := ^ j∈I ^ k∈I (^ i∈N (pij →pik)∧ _ s∈N (¬psj ∧psk))→(pj →pk) .
First, we quantify over all the possible choices of two issues j and k. The antecedent inside the parentheses is going to be true if and only if all the agents who accept j also